In many of the predictive and analytical problems of theoretical and applied engineering, it is necessary to solve one or more of the equations of physics that describe a physical field. The field variable that is sought may be, for example, a measure of mechanical deformation, acoustic pressure, an electrostatic or electromagnetic potential, or an electromagnetic field intensity.
The equations that describe these and other fields are differential equations, integral equations, or equations that combine aspects of both, which can be solved exactly, if at all, only by the techniques of calculus. Typically, boundary conditions (and also initial conditions, if time evolution is part of the problem to be solved) must be specified in order to provide a complete definition of the problem that is to be solved. In practical situations, these boundary conditions often involve specifying the field variable, its derivatives, functions of either or both of these, or a combination of some or all of the preceding, on complicated two- or three-dimensional surfaces. As a consequence of the complexity of both the governing physical equations and the boundary conditions, most practical problems are too complex to solve by hand calculations.
Numerical modeling techniques are aimed at providing approximate solutions to these equations that can be performed with the help of a digital computer. Many different numerical modeling techniques are available. One such technique, which has proven extremely fruitful in numerous fields of application, is the so-called finite element method (FEM).
As we describe more fully hereafter, the traditional practice of the FEM calls for the spatial region of interest to be subdivided into a plurality of perfectly juxtaposed cells (i.e., cells that are juxtaposed without overlaps or interstitial voids), which are referred to as elements. Discrete nodes are defined on the inter-element boundaries and, typically, also within the element interiors. The network of elements and nodes is referred to as a mesh.
The FEM does not seek an exact solution to the physical field equation. Instead, it assumes that within each element, the field can be described, to an adequate approximation, by a finite linear combination of simple functions, such as polynomials, that are chosen, inter alia, for convenient analytical properties. The coefficients of the respective polynomials in such a linear combination are referred to as degrees of freedom (DOF). The polynomials are normalized in such a way that the DOF are equal to, or otherwise relate to, the values of the field variable at the nodes. Between the nodes, the polynomials interpolate the values of the field variable in a continuous fashion.
The polynomials themselves are known a priori. Therefore, the problem is solved, within a given element, by specifying the values of the DOF for that element. The DOF are not determined directly from the field equation. Instead, a mathematical condition is derived from the field equation. In essence, this condition demands that a certain measure of error (i.e., between the approximate and exact solutions) must be small. Once this condition has been fully defined, it can be expressed as a set of linear equations, in which the unknowns are the DOF. Linear equations are well-suited to be solved by a digital computer, because all that is required is a large number of repetitive, mechanical manipulations of stored quantities.
The linear equations belonging to a given element are not solved in isolation. Instead, the sets of equations belonging to all of the elements are assembled into a single matrix system. Suitable modifications are made to these system equations to account for the boundary conditions. Then the matrix system is solved automatically by standard methods.
Special challenges arise when attempting to obtain a mathematical solution, either exact or approximate, to any problem involving scattering and/or radiation, e.g., of acoustic or electromagnetic waves, from an object located in an unbounded region (that is, in open or free space). The mathematical solution to all such problems must satisfy a so-called "radiation condition" (often known as the Sommerfeld condition, or, in electromagnetism, in acoustics and as the Silver-Muller condition. The condition states that all waves "at infinity" are only traveling outward toward infinity, not inward from infinity. Thus, all the energy in the problem resides in the radiated or scattered waves, which are traveling outward after their interaction with the object; conversely, no energy is created at infinity. Note that the radiation condition is a condition that exists "at infinity," not at a finite distance. Much of the history of computational methods for such problems has been focused on how to obtain approximate, numerical solutions that satisfy the radiation condition to an acceptable accuracy, while not being prohibitively expensive.
We note that the FEM, because of its extraordinary versatility in handling objects of virtually any geometric shape and material properties, is generally the method of choice for modeling the finite part of the problem, i.e., the object and, sometimes, a finite part of the open region surrounding the object. (The Finite Difference Time Domain method is occasionally used for small problems.) The challenge is how to model the remainder of the infinite region, including the radiation condition at infinity. The methods have generally fallen into three classes.
The first class uses boundary integral equation methods (BIEM). Here, an integral equation that satisfies the radiation condition exactly can be applied directly on the outer surface of the object. Its advantage, which seems compelling at first sight, is that the infinite exterior domain is replaced, with no loss in physical approximation, by a (relatively) small surface, which greatly reduces the computational size of the problem. However, it has a severe disadvantage. The matrices in the resulting discretization of the integral equation are fully populated, making the computational cost prohibitively expensive except for only small-scale problems. (Small-scale means the dimensions of the object are small relative to the relevant wavelengths.)
The second class uses exact solutions to the wave equation in open regions, often expressed as infinite series of known functions, e.g., wave functions or multipoles. These are joined to the solutions in the finite region in a manner that approximately establishes continuity along, e.g., a closed boundary surrounding the object. This approach suffers from the same disadvantage as the BIEM, namely, that the resulting discretized equations are fully populated, hence prohibitively expensive.
The third class, which has comprised most of the research in recent years, is to construct an artificial boundary surrounding the object, then apply a so-called absorbing (or non-reflecting) boundary condition (ABC) that will make the boundary appear as transparent as possible to all outward traveling waves, i.e., the radiated or scattered fields. There have been many variations on this approach. The primary advantage here is that the resulting discretized equations have sparse matrices. This property, by itself, would keep the cost low. However, the disadvantage is that all ABCs, which are applied to a boundary at a finite distance, can only be approximations to the exact radiation condition at infinity. As a consequence, spurious (non-realistic) waves are reflected from the artificial boundary, which then propagate throughout the entire finite region, contaminating the solution everywhere. This effect can be mitigated by moving the artificial boundary farther away from the object, but this only increases the size of the finite region and hence the cost.
An alternative to these approaches is the use of so-called infinite elements, which are finite elements that cover a semi-infinite sector of space. (We note that the term "finite" in the name "finite element method" means non-infinitesimal; it does not mean non-infinite. It is therefore semantically correct to say "infinite finite element," as was done in the early 1970s, but it is now universally called an "infinite element."
Infinite elements have been used to great advantage to solve problems in acoustics. Such uses for infinite elements are described, for example, in the article by D. S. Burnett, "A Three-Dimensional Acoustic Infinite Element Based on a Prolate Spheroidal Multipole Expansion," J. Acoust. Soc. Am. 96 (1994) 2798-2816 (BURNETT 1994). Also pertinent in this regard are U.S. Pat. No. 5,604,891 and U.S. Pat. No. 5,604,893, both assigned to the assignee hereof. Also pertinent in this regard is the currently pending U.S. patent application Ser. No. 08/812,472, also assigned to the assignee hereof.
These acoustic infinite elements have been highly acclaimed by leaders in the academic community as well as by commercial code developers because they have exhibited both high accuracy and extraordinary speed of computation (over 400 times faster than other state-of-the-art methods).
To use these infinite elements, one constructs (similar to the third class of methods above) an artificial boundary surrounding the object. However, instead of using ABCs, one constructs a single layer of infinite elements around the entire artificial boundary. This single layer covers the entire infinite region outside the artificial boundary. Most important to the invention is the fact that because each element extends all the way to infinity, the essential radiation condition can now be applied exactly "at" infinity. This is primarily what accounts for the high accuracy of these infinite elements. This high accuracy, in turn, accounts for the extremely high computational speeds because the infinite elements can be placed extraordinarily close to the object (typically less than half a wavelength), resulting in a much smaller finite computational region (than required with ABCs). Below, we describe an electromagnetic infinite element. We are unaware of any prior use of infinite elements in electromagnetism.
A central challenge to developing an EM infinite element is related to the vector nature of EM fields (as opposed to the simpler scalar nature of acoustic fields). Thus, an EM field is characterized by two coupled vector fields, i.e., the electric field and the magnetic field. Although all the classical field theories permit the representation of (scalar or vector) fields as spatial derivatives of potential functions, in the case of EM the only known representation for general applications (i.e., inhomogeneous and/or anisotropic physical properties) involves a vector-valued potential function, known as the vector potential. The lack of a suitable scalar-valued potential has been a deterrent to the development of an EM infinite element.